Theorem equsex

Description: An equivalence related to implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 . See equsexvw and equsexv for versions with disjoint variable conditions proved from fewer axioms. See also the dual form equsal . See equsexALT for an alternate proof. (Contributed by NM, 5-Aug-1993) (Revised by Mario Carneiro, 3-Oct-2016) (Proof shortened by Wolf Lammen, 6-Feb-2018) (New usage is discouraged.)

	Ref	Expression
Hypotheses	equsal.1	⊢ Ⅎ 𝑥 𝜓
	equsal.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
Assertion	equsex	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		equsal.1	⊢ Ⅎ 𝑥 𝜓
2		equsal.2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ 𝜓 ) )
3	2	biimpa	⊢ ( ( 𝑥 = 𝑦 ∧ 𝜑 ) → 𝜓 )
4	1 3	exlimi	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) → 𝜓 )
5	1 2	equsal	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ↔ 𝜓 )
6		equs4	⊢ ( ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) )
7	5 6	sylbir	⊢ ( 𝜓 → ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) )
8	4 7	impbii	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ 𝜑 ) ↔ 𝜓 )